On the length of an external branch in the Beta-coalescent

In this paper, we consider Beta ( 2 − α , α ) (with 1 < α < 2 ) and related Λ -coalescents. If T ( n ) denotes the length of a randomly chosen external branch of the n -coalescent, we prove the convergence of n α − 1 T ( n ) when n tends to ∞ , and give the limit. To this aim, we give asymptotics for the number σ ( n ) of collisions which occur in the n -coalescent until the end of the chosen external branch, and for the block counting process associated with the n -coalescent.